Density Module¶
Overview
Computes the complex pairwise correlation function. 

Computes the density of a system on a grid. 

Computes the local density around a particle. 

Computes the RDF \(g \left( r \right)\) for supplied data. 

Computes a grid of voxels occupied by spheres. 
Details
The freud.density
module contains various classes relating to the
density of the system. These functions allow evaluation of particle
distributions with respect to other particles.

class
freud.density.
CorrelationFunction
¶ Bases:
freud.locality._SpatialHistogram1D
Computes the complex pairwise correlation function.
The correlation function is given by \(C(r) = \left\langle s^*_1(0) \cdot s_2(r) \right\rangle\) between two sets of points \(p_1\) (
points
) and \(p_2\) (query_points
) with associated values \(s_1\) (values
) and \(s_2\) (query_values
). Computing the correlation function results in an array of the expected (average) product of all values at a given radial distance \(r\). The values of \(r\) where the correlation function is computed are controlled by thebins
andr_max
parameters to the constructor, and the spacing between the bins is given bydr = r_max / bins
.Note
Selfcorrelation: It is often the case that we wish to compute the correlation function of a set of points with itself. If
query_points
is the same aspoints
, not provided, orNone
, we omit accumulating the selfcorrelation value in the first bin. Parameters
bins (unsigned int) – The number of bins in the correlation function.
r_max (float) – Maximum pointwise distance to include in the calculation.

bin_centers
¶ The centers of each bin in the histogram.
 Type
\((N_{bins}, )\)
numpy.ndarray

property
bin_counts
¶ The bin counts in the histogram.
 Type

bin_edges
¶ The edges of each bin in the histogram. It is one element larger because each bin has a lower and an upper bound.
 Type
\((N_{bins}+1, )\)
numpy.ndarray

property
box
¶ The box object used in the last computation.
 Type

compute
(self, system, values, query_points=None, query_values=None, neighbors=None, reset=True)¶ Calculates the correlation function and adds to the current histogram.
 Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.values ((\(N_{points}\))
numpy.ndarray
) – Values associated with the system points used to calculate the correlation function.query_points ((\(N_{query\_points}\), 3)
numpy.ndarray
, optional) – Query points used to calculate the correlation function. Uses the system’s points ifNone
(Default value =None
).query_values ((\(N_{query\_points}\))
numpy.ndarray
, optional) – Query values used to calculate the correlation function. Usesvalues
ifNone
. (Default value =None
).neighbors (
freud.locality.NeighborList
or dict, optional) – Either aNeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).reset (bool) – Whether to erase the previously computed values before adding the new computation; if False, will accumulate data (Default value: True).

property
correlation
¶ Expected (average) product of all values at a given radial distance.
 Type
(\(N_{bins}\))
numpy.ndarray

default_query_args
¶ The default query arguments are
{'mode': 'ball', 'r_max': self.r_max}
.

plot
(self, ax=None)¶ Plot complex correlation function.
 Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
) Returns
Axis with the plot.
 Return type

class
freud.density.
GaussianDensity
¶ Bases:
freud.util._Compute
Computes the density of a system on a grid.
Replaces particle positions with a Gaussian blur and calculates the contribution from each to the proscribed grid based upon the distance of the grid cell from the center of the Gaussian. The weights for the Gaussians could be additionally specified in the compute method. The convolution of the weights with the Gaussians is calculated in this case:
\[p(\vec{r}) = \sum_i \frac{1}{2\pi \sigma^2} \exp \left(\frac{(\vec{r}\vec{r}_i)^2}{2\sigma^2}\right) p_i\]The resulting data is a regular grid of particle densities or convolved parameter that can be used in standard algorithms requiring evenly spaced point, such as Fast Fourier Transforms. The dimensions of the grid are set in the constructor, and can either be set equally for all dimensions or for each dimension independently.
 Parameters

property
box
¶ Box used in the calculation.
 Type

compute
(self, system, values=None)¶ Calculates the Gaussian blur for the specified points.
 Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.values ((\(N_{points}\))
numpy.ndarray
) – Values associated with the system points used to calculate the convolution. Calculates Gaussian blur (equivalent to providing a value of 1 for every point) ifNone
. (Default value =None
).

property
density
¶ The grid with the Gaussian density contributions from each point.
 Type
(\(w_x\), \(w_y\), \(w_z\))
numpy.ndarray

plot
(self, ax=None)¶ Plot Gaussian Density.
 Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
) Returns
Axis with the plot.
 Return type

class
freud.density.
LocalDensity
¶ Bases:
freud.locality._PairCompute
Computes the local density around a particle.
The density of the local environment is computed and averaged for a given set of query points in a sea of data points. Providing the same points calculates them against themselves. Computing the local density results in an array listing the value of the local density around each query point. Also available is the number of neighbors for each query point, giving the user the ability to count the number of particles in that region. Note that the computed density is essentially a number density (allowing for fractional values as described below). If particles have a specific volume, a volume density can be computed by simply multiplying the number density by the volume of the particles.
In order to provide sufficiently smooth data, data points can be fractionally counted towards the density. Rather than perform computeintensive area (volume) overlap calculations to determine the exact amount of overlap area (volume), the LocalDensity class performs a simple linear interpolation relative to the centers of the data points. Specifically, a point is counted as one neighbor of a given query point if it is entirely contained within the
r_max
, half of a neighbor if the distance to its center is exactlyr_max
, and zero if its center is a distance greater than or equal tor_max + diameter
from the query point’s center. Graphically, this looks like: Parameters

property
box
¶ Box used in the calculation.
 Type

compute
(self, system, query_points=None, neighbors=None)¶ Calculates the local density for the specified points.
Example:
>>> import freud >>> box, points = freud.data.make_random_system(10, 100, seed=0) >>> # Compute Local Density >>> ld = freud.density.LocalDensity(r_max=3, diameter=0.05) >>> ld.compute(system=(box, points)) freud.density.LocalDensity(...)
 Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.query_points ((\(N_{query\_points}\), 3)
numpy.ndarray
, optional) – Query points used to calculate the correlation function. Uses the system’s points ifNone
(Default value =None
).neighbors (
freud.locality.NeighborList
or dict, optional) –Either a
NeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).

default_query_args
¶ The default query arguments are
{'mode': 'ball', 'r_max': self.r_max + 0.5*self.diameter}
.

property
density
¶ Density of points per query point.
 Type
(\(N_{points}\))
numpy.ndarray

property
num_neighbors
¶ Number of neighbor points for each query point.
 Type
(\(N_{points}\))
numpy.ndarray

class
freud.density.
RDF
¶ Bases:
freud.locality._SpatialHistogram1D
Computes the RDF \(g \left( r \right)\) for supplied data.
Note that the RDF is defined strictly according to the pair correlation function, i.e.
\[g(r) = V\frac{N1}{N} \langle \delta(r) \rangle\]In the thermodynamic limit, the fraction tends to unity and the limiting behavior of \(\lim_{r \to \infty} g(r)=1\) is recovered. However, for very small systems the long range behavior of the radial distribution will instead tend to \(\frac{N1}{N}\). In small systems, where this deviation is noticeable, the
normalize
flag may be used to rescale the results and force the long range behavior to 1. Note that this option will have little to no effect on larger systems (for example, for systems of 100 particles the RDF will differ by 1%).Note
2D:
freud.density.RDF
properly handles 2D boxes. The points must be passed in as[x, y, 0]
. Parameters
bins (unsigned int) – The number of bins in the RDF.
r_max (float) – Maximum interparticle distance to include in the calculation.
r_min (float, optional) – Minimum interparticle distance to include in the calculation (Default value =
0
).normalize (bool, optional) – Scale the RDF values by \(\frac{N_{query\_points}}{N_{query\_points}1}\). This argument primarily exists to deal with standard RDF calculations where no special
query_points
orneighbors
are provided, but where the number ofquery_points
is small enough that the longranged limit of \(g(r)\) deviates significantly from \(1\). It should not be used ifquery_points
is provided as a different set of points, or if unusual query arguments are provided tocompute()
, specifically ifexclude_ii
is set toFalse
. This normalization is not meaningful in such cases and will simply convolute the data.

bin_centers
¶ The centers of each bin in the histogram.
 Type
\((N_{bins}, )\)
numpy.ndarray

property
bin_counts
¶ The bin counts in the histogram.
 Type

bin_edges
¶ The edges of each bin in the histogram. It is one element larger because each bin has a lower and an upper bound.
 Type
\((N_{bins}+1, )\)
numpy.ndarray

property
box
¶ The box object used in the last computation.
 Type

compute
(self, system, query_points=None, neighbors=None, reset=True)¶ Calculates the RDF and adds to the current RDF histogram.
 Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.query_points ((\(N_{query\_points}\), 3)
numpy.ndarray
, optional) – Query points used to calculate the RDF. Uses the system’s points ifNone
(Default value =None
).neighbors (
freud.locality.NeighborList
or dict, optional) –Either a
NeighborList
of neighbor pairs to use in the calculation, or a dictionary of query arguments (Default value: None).reset (bool) – Whether to erase the previously computed values before adding the new computation; if False, will accumulate data (Default value: True).

default_query_args
¶ The default query arguments are
{'mode': 'ball', 'r_max': self.r_max}
.

property
n_r
¶ Histogram of cumulative bin_counts values. More precisely,
n_r[i]
is the average number of points contained within a ball of radiusbin_edges[i+1]
centered at a givenquery_point
averaged over allquery_points
in the last call tocompute()
. Type
(\(N_{bins}\),)
numpy.ndarray

plot
(self, ax=None)¶ Plot radial distribution function.
 Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
) Returns
Axis with the plot.
 Return type

property
rdf
¶ Histogram of RDF values.
 Type
(\(N_{bins}\),)
numpy.ndarray

class
freud.density.
SphereVoxelization
¶ Bases:
freud.util._Compute
Computes a grid of voxels occupied by spheres.
This class constructs a grid of voxels. From a given set of points and a desired radius, a set of spheres are created. The voxels are assigned a value of 1 if their center is contained in one or more spheres and 0 otherwise. The dimensions of the grid are set in the constructor, and can either be set equally for all dimensions or for each dimension independently.
 Parameters

property
box
¶ Box used in the calculation.
 Type

compute
(self, system)¶ Calculates the voxelization of spheres about the specified points.
 Parameters
system – Any object that is a valid argument to
freud.locality.NeighborQuery.from_system
.

plot
(self, ax=None)¶ Plot voxelization.
 Parameters
ax (
matplotlib.axes.Axes
, optional) – Axis to plot on. IfNone
, make a new figure and axis. (Default value =None
) Returns
Axis with the plot.
 Return type

property
voxels
¶ The voxel grid indicating overlap with the computed spheres.
 Type
(\(w_x\), \(w_y\), \(w_z\))
numpy.ndarray